Let $G$ be a compact group and let $f_{ij}\in L_2(G)$ be bandlimited functions. We define the Non-Unique Games (NUG) problem as finding $g_1\ldots,g_n\in G$ to minimize $\sum_{i,j=1}^n f_{ij}(g_i g_j^{-1})$. We devise a relaxation of the NUG problem to a semidefinite program (SDP) by taking the Fourier transform of $f_{ij}$ over $G$, which can then be solved efficiently.

In the matrix completion problem, we have a matrix $M$ where we are only given a small number of its entries and our goal is to fill in the rest of the entries. While this problem is impossible to solve for general matrices, it can be solved if $M$ has additional structure, such as being low rank. In this talk, I will describe how the matrix completion problem can be solved by nuclear norm minimization and how this can be generalized to tensor completion via the sum of squares hierarchy.

Advanced LIGO discovered cosmic gravitational waves and surprised us with giant binary black-hole systems, just in time for the 100th anniversary of Einstein's prediction. Gravitational waves became the latest window on the Universe from violent transients to cosmology. I will discuss some aspects of (i) the instrumental breakthroughs that enabled the unprecedented sensitivity reached by Advanced LIGO and (ii) the key scientific directions in which gravitational wave searches are being utilized in the context of multimessenger astronomy focusing on the future.